Theorem sge0tsms 43808

Description: Σ^{^} applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0tsms.g	⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))
sge0tsms.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
sge0tsms.f	⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))

Assertion

Ref	Expression
sge0tsms	⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))

Proof of Theorem sge0tsms

Dummy variables 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )
2	1	a1i 11	. . 3 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
3		xrltso 12804	. . . . . 6 ⊢ < Or ℝ*
4	3	supex 9152	. . . . 5 ⊢ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V)
6		elsng 4572	. . . 4 ⊢ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )))
8	2, 7	mpbird 256	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )})
9		sge0tsms.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
11		sge0tsms.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
14	10, 12, 13	sge0pnfval 43801	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
15	11	ffnd 6585	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝑋)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹 Fn 𝑋)
17		fvelrnb 6812	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
18	16, 17	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
19	13, 18	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞)
20		iccssxr 13091	. . . . . . . . . . . . . 14 ⊢ (0[,]+∞) ⊆ ℝ*
21		sge0tsms.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))
22		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
23	11	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
24		elinel1 4125	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
25		elpwi 4539	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
26	24, 25	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
27	26	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
28		fssres 6624	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 ⊆ 𝑋) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
29	23, 27, 28	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
30		elinel2 4126	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
31	30	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
32		0red 10909	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
33	29, 31, 32	fdmfifsupp 9068	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) finSupp 0)
34	21, 22, 29, 33	gsumge0cl 43799	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) ∈ (0[,]+∞))
35	20, 34	sselid 3915	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
36	35	ralrimiva 3107	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
37	36	3ad2ant1 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
38		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥)))
39	38	rnmptss 6978	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ*)
40	37, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ*)
41		snelpwi 5354	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ 𝒫 𝑋)
42		snfi 8788	. . . . . . . . . . . . . . 15 ⊢ {𝑦} ∈ Fin
43	42	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ Fin)
44	41, 43	elind 4124	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
45	44	3ad2ant2 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
46	11	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
47		snssi 4738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ⊆ 𝑋)
48	47	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ 𝑋)
49	46, 48	fssresd 6625	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
50	49	feqmptd 6819	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
51		fvres 6775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹‘𝑥))
52	51	mpteq2ia 5173	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))
53	52	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥)))
54	50, 53	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥)))
55	54	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))))
56	55	3adant3 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))))
57		xrge0cmn 20552	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd
58	21, 57	eqeltri 2835	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ∈ CMnd
59		cmnmnd 19317	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
60	58, 59	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝐺 ∈ Mnd
61	60	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝐺 ∈ Mnd)
62		simp2 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝑦 ∈ 𝑋)
63	11	ffvelrnda 6943	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ (0[,]+∞))
64	63	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) ∈ (0[,]+∞))
65		df-ss 3900	. . . . . . . . . . . . . . . . . 18 ⊢ ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
66	20, 65	mpbi 229	. . . . . . . . . . . . . . . . 17 ⊢ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
67	66	eqcomi 2747	. . . . . . . . . . . . . . . 16 ⊢ (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
68		ovex 7288	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]+∞) ∈ V
69		xrsbas 20526	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ* = (Base‘ℝ*𝑠)
70	21, 69	ressbas 16873	. . . . . . . . . . . . . . . . 17 ⊢ ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
71	68, 70	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
72	67, 71	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) = (Base‘𝐺)
73		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
74	72, 73	gsumsn 19470	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))) = (𝐹‘𝑦))
75	61, 62, 64, 74	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))) = (𝐹‘𝑦))
76		simp3 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) = +∞)
77	56, 75, 76	3eqtrrd 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
78		reseq2 5875	. . . . . . . . . . . . . 14 ⊢ (𝑥 = {𝑦} → (𝐹 ↾ 𝑥) = (𝐹 ↾ {𝑦}))
79	78	oveq2d 7271	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑦} → (𝐺 Σg (𝐹 ↾ 𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
80	79	rspceeqv 3567	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥)))
81	45, 77, 80	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥)))
82		pnfxr 10960	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ*
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ℝ*)
84	38	elrnmpt 5854	. . . . . . . . . . . 12 ⊢ (+∞ ∈ ℝ* → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥))))
85	83, 84	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥))))
86	81, 85	mpbird 256	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
87		supxrpnf 12981	. . . . . . . . . 10 ⊢ ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
88	40, 86, 87	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
89	88	3exp 1117	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)))
90	89	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)))
91	90	rexlimdv 3211	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞))
92	19, 91	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
93	14, 92	eqtr4d 2781	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
94	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
95	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
96		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
97	95, 96	fge0iccico 43798	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
98	94, 97	sge0reval 43800	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
99	23, 27	feqresmpt 6820	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
100	99	adantlr 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
101	100	oveq2d 7271	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) = (𝐺 Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
102	21	fveq2i 6759	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
103		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞))
104		xrsadd 20527	. . . . . . . . . . . . . 14 ⊢ +𝑒 = (+g‘ℝ*𝑠)
105	103, 104	ressplusg 16926	. . . . . . . . . . . . 13 ⊢ ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))))
106	68, 105	ax-mp 5	. . . . . . . . . . . 12 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
107	106	eqcomi 2747	. . . . . . . . . . 11 ⊢ (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) = +𝑒
108	102, 107	eqtr2i 2767	. . . . . . . . . 10 ⊢ +𝑒 = (+g‘𝐺)
109	21	oveq1i 7265	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (0[,)+∞)) = ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞))
110		icossicc 13097	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
111	68, 110	pm3.2i 470	. . . . . . . . . . . 12 ⊢ ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
112		ressabs 16885	. . . . . . . . . . . 12 ⊢ (((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞)))
113	111, 112	ax-mp 5	. . . . . . . . . . 11 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞))
114	109, 113	eqtr2i 2767	. . . . . . . . . 10 ⊢ (ℝ*𝑠 ↾s (0[,)+∞)) = (𝐺 ↾s (0[,)+∞))
115	58	elexi 3441	. . . . . . . . . . 11 ⊢ 𝐺 ∈ V
116	115	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺 ∈ V)
117		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
118	110	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ⊆ (0[,]+∞))
119		0xr 10953	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ*
120	119	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 0 ∈ ℝ*)
121	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → +∞ ∈ ℝ*)
122	95	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,]+∞))
123	26	sselda 3917	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
124	123	adantll 710	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
125	122, 124	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
126	20, 125	sselid 3915	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ*)
127		iccgelb 13064	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹‘𝑦))
128	120, 121, 125, 127	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 0 ≤ (𝐹‘𝑦))
129		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑦) = +∞ → (𝐹‘𝑦) = +∞)
130	129	eqcomd 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = +∞ → +∞ = (𝐹‘𝑦))
131	130	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ = (𝐹‘𝑦))
132	11	ffund 6588	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → Fun 𝐹)
133	132	ad2antrr 722	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → Fun 𝐹)
134	22, 123	sylan 579	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
135	11	fdmd 6595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom 𝐹 = 𝑋)
136	135	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋 = dom 𝐹)
137	136	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑋 = dom 𝐹)
138	134, 137	eleqtrd 2841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
139		fvelrn 6936	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ ran 𝐹)
140	133, 138, 139	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ran 𝐹)
141	140	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) ∈ ran 𝐹)
142	131, 141	eqeltrd 2839	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran 𝐹)
143	142	adantl3r 746	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran 𝐹)
144	96	ad3antrrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
145	143, 144	pm2.65da 813	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑦) = +∞)
146	145	neqned 2949	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ≠ +∞)
147		ge0xrre 42959	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) ∈ (0[,]+∞) ∧ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ℝ)
148	125, 146, 147	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
149	148	ltpnfd 12786	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) < +∞)
150	120, 121, 126, 128, 149	elicod 13058	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞))
151		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))
152	150, 151	fmptd 6970	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)):𝑥⟶(0[,)+∞))
153		0e0icopnf 13119	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,)+∞)
154	153	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
155		eliccxr 13096	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
156		xaddid2 12905	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
157		xaddid1 12904	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (𝑦 +𝑒 0) = 𝑦)
158	156, 157	jca 511	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
159	155, 158	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,]+∞) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
160	159	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ (0[,]+∞)) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
161	72, 108, 114, 116, 117, 118, 152, 154, 160	gsumress 18281	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
162		rege0subm 20566	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
163	162	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
164		eqid 2738	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (0[,)+∞)) = (ℂfld ↾s (0[,)+∞))
165	117, 163, 152, 164	gsumsubm 18388	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
166		eqidd 2739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
167		vex 3426	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
168	167	mptex 7081	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ∈ V
169	168	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ∈ V)
170		ovexd 7290	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld ↾s (0[,)+∞)) ∈ V)
171		ovexd 7290	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠 ↾s (0[,)+∞)) ∈ V)
172		rge0ssre 13117	. . . . . . . . . . . . . . . . 17 ⊢ (0[,)+∞) ⊆ ℝ
173		ax-resscn 10859	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
174	172, 173	sstri 3926	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℂ
175		cnfldbas 20514	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = (Base‘ℂfld)
176	164, 175	ressbas2 16875	. . . . . . . . . . . . . . . 16 ⊢ ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂfld ↾s (0[,)+∞))))
177	174, 176	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) = (Base‘(ℂfld ↾s (0[,)+∞)))
178	177	eqcomi 2747	. . . . . . . . . . . . . 14 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (0[,)+∞)
179	110, 20	sstri 3926	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ ℝ*
180		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (ℝ*𝑠 ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞))
181	180, 69	ressbas2 16875	. . . . . . . . . . . . . . 15 ⊢ ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞))))
182	179, 181	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞)))
183	178, 182	eqtri 2766	. . . . . . . . . . . . 13 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞)))
184	183	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞))))
185		rge0srg 20581	. . . . . . . . . . . . . . 15 ⊢ (ℂfld ↾s (0[,)+∞)) ∈ SRing
186	185	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (ℂfld ↾s (0[,)+∞)) ∈ SRing)
187		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
188		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
189		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℂfld ↾s (0[,)+∞)))
190		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = (+g‘(ℂfld ↾s (0[,)+∞)))
191	189, 190	srgacl 19675	. . . . . . . . . . . . . 14 ⊢ (((ℂfld ↾s (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
192	186, 187, 188, 191	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
193	192	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
194	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
195		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
196	195, 178	eleqtrdi 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
197	194, 196	sseldd 3918	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ ℝ)
198	197	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑠 ∈ ℝ)
199	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
200		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
201	200, 178	eleqtrdi 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
202	199, 201	sseldd 3918	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ ℝ)
203	202	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑡 ∈ ℝ)
204		rexadd 12895	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
205	204	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
206	162	elexi 3441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0[,)+∞) ∈ V
207		cnfldadd 20515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ + = (+g‘ℂfld)
208	164, 207	ressplusg 16926	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0[,)+∞) ∈ V → + = (+g‘(ℂfld ↾s (0[,)+∞))))
209	206, 208	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ + = (+g‘(ℂfld ↾s (0[,)+∞)))
210	209, 207	eqtr3i 2768	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = (+g‘ℂfld)
211	210, 207	eqtr4i 2769	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = +
212	211	oveqi 7268	. . . . . . . . . . . . . . . 16 ⊢ (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
213	212	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
214	180, 104	ressplusg 16926	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,)+∞))))
215	206, 214	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,)+∞)))
216	215	eqcomi 2747	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘(ℝ*𝑠 ↾s (0[,)+∞))) = +𝑒
217	216	oveqi 7268	. . . . . . . . . . . . . . . 16 ⊢ (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
218	217	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
219	205, 213, 218	3eqtr4d 2788	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
220	198, 203, 219	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
221	220	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
222		funmpt 6456	. . . . . . . . . . . . 13 ⊢ Fun (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))
223	222	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
224	150, 177	eleqtrdi 2849	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
225	224	ralrimiva 3107	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
226	151	rnmptss 6978	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → ran (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ⊆ (Base‘(ℂfld ↾s (0[,)+∞))))
227	225, 226	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ⊆ (Base‘(ℂfld ↾s (0[,)+∞))))
228	169, 170, 171, 184, 193, 221, 223, 227	gsumpropd2 18279	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
229	165, 166, 228	3eqtrd 2782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
230	30	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
231	148	recnd 10934	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℂ)
232	230, 231	gsumfsum 20577	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
233	229, 232	eqtr3d 2780	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
234	101, 161, 233	3eqtrrd 2783	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (𝐺 Σg (𝐹 ↾ 𝑥)))
235	234	mpteq2dva 5170	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
236	235	rneqd 5836	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
237	236	supeq1d 9135	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
238	98, 237	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
239	93, 238	pm2.61dan 809	. . 3 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
240	21, 9, 11, 1	xrge0tsms 23903	. . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )})
241	239, 240	eleq12d 2833	. 2 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ (𝐺 tsums 𝐹) ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )}))
242	8, 241	mpbird 256	1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   ↾ cres 5582  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  supcsup 9129  ℂcc 10800  ℝcr 10801  0cc0 10802   + caddc 10805  +∞cpnf 10937  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   +_𝑒 cxad 12775  [,)cico 13010  [,]cicc 13011  Σcsu 15325  Basecbs 16840   ↾_s cress 16867  +_gcplusg 16888   Σ_g cgsu 17068  ℝ^*_𝑠cxrs 17128  Mndcmnd 18300  SubMndcsubmnd 18344  CMndccmn 19301  SRingcsrg 19656  ℂ_fldccnfld 20510   tsums ctsu 23185  Σ^{^}csumge0 43790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xadd 12778  df-ioo 13012  df-ioc 13013  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-ordt 17129  df-xrs 17130  df-mre 17212  df-mrc 17213  df-acs 17215  df-ps 18199  df-tsr 18200  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-srg 19657  df-ring 19700  df-cring 19701  df-fbas 20507  df-fg 20508  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-ntr 22079  df-nei 22157  df-cn 22286  df-haus 22374  df-fil 22905  df-fm 22997  df-flim 22998  df-flf 22999  df-tsms 23186  df-sumge0 43791

This theorem is referenced by: (None)